J-PAKE: Password-Authenticated Key Exchange by Juggling

Abstract

This document specifies a Password-Authenticated Key Exchange by
Juggling (J-PAKE) protocol. This protocol allows the establishment
of a secure end-to-end communication channel between two remote
parties over an insecure network solely based on a shared password,
without requiring a Public Key Infrastructure (PKI) or any trusted
third party.

Status of This Memo

This document is not an Internet Standards Track specification; it is
published for informational purposes.

This is a contribution to the RFC Series, independently of any other
RFC stream. The RFC Editor has chosen to publish this document at
its discretion and makes no statement about its value for
implementation or deployment. Documents approved for publication by
the RFC Editor are not a candidate for any level of Internet
Standard; see Section 2 of RFC 7841.

Information about the current status of this document, any errata,
and how to provide feedback on it may be obtained at
http://www.rfc-editor.org/info/rfc8236.

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1. Introduction

Password-Authenticated Key Exchange (PAKE) is a technique that aims
to establish secure communication between two remote parties solely
based on their shared password, without relying on a Public Key
Infrastructure or any trusted third party [BM92]. The first PAKE
protocol, called Encrypted Key Exchange (EKE), was proposed by Steven
Bellovin and Michael Merrit in 1992 [BM92]. Other well-known PAKE
protocols include Simple Password Exponential Key Exchange (SPEKE) by
David Jablon in 1996 [Jab96] and Secure Remote Password (SRP) by Tom
Wu in 1998 [Wu98]. SRP has been revised several times to address
reported security and efficiency issues. In particular, the version
6 of SRP, commonly known as SRP-6, is specified in [RFC5054].

This document specifies a PAKE protocol called Password-Authenticated
Key Exchange by Juggling (J-PAKE), which was designed by Feng Hao and
Peter Ryan in 2008 [HR08]. There are a few factors that may be
considered in favor of J-PAKE. First, J-PAKE has security proofs,
while equivalent proofs are lacking in EKE, SPEKE and SRP-6. Second,
J-PAKE follows a completely different design approach from all other
PAKE protocols, and is built upon a well-established Zero Knowledge
Proof (ZKP) primitive: Schnorr NIZK proof [RFC8235]. Third, J-PAKE
adopts novel engineering techniques to optimize the use of ZKP so
that overall the protocol is sufficiently efficient for practical
use. Fourth, J-PAKE is designed to work generically in both the
finite field and elliptic curve settings (i.e., DSA and ECDSA-like
groups, respectively). Unlike SPEKE, it does not require any extra
primitive to hash passwords onto a designated elliptic curve. Unlike
SPAKE2 [AP05] and SESPAKE [SOAA15], it does not require a trusted
setup (i.e., the so-called common reference model) to define a pair
of generators whose discrete logarithm must be unknown. Finally,
J-PAKE has been used in real-world applications at a relatively large
scale, e.g., Firefox sync [MOZILLA], Pale moon sync [PALEMOON], and
Google Nest products [ABM15]. It has been included into widely
distributed open source libraries such as OpenSSL [BOINC], Network
Security Services (NSS) [MOZILLA_NSS], and the Bouncy Castle
[BOUNCY]. Since 2015, J-PAKE has been included in Thread [THREAD] as
a standard key agreement mechanism for IoT (Internet of Things)
applications, and also included in ISO/IEC 11770-4:2017
[ISO.11770-4].

1.1. Requirements Language

The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
"SHOULD", "SHOULD NOT", "RECOMMENDED", "NOT RECOMMENDED", "MAY", and
"OPTIONAL" in this document are to be interpreted as described in
BCP 14 [RFC2119] [RFC8174] when, and only when, they appear in all
capitals, as shown here.

1.2. Notation

The following notation is used in this document:

Alice: the assumed identity of the prover in the protocol

Bob: the assumed identity of the verifier in the protocol

s: a low-entropy secret shared between Alice and Bob

o a | b: a divides b
o a || b: concatenation of a and b

[a, b]: the interval of integers between and including a and b

H: a secure cryptographic hash function

o p: a large prime
o q: a large prime divisor of p-1, i.e., q | p-1

Zp*: a multiplicative group of integers modulo p

Gq: a subgroup of Zp* with prime order q

o g: a generator of Gq
o g^d: g raised to the power of d
o a mod b: a modulo b

Fp: a finite field of p elements, where p is a prime

E(Fp): an elliptic curve defined over Fp

G: a generator of the subgroup over E(Fp) with prime order n

o n: the order of G

h: the cofactor of the subgroup generated by G, which is equal to
the order of the elliptic curve divided by n

P x [b]: multiplication of a point P with a scalar b over E(Fp)

KDF(a): Key Derivation Function with input a

MAC(MacKey, MacData): MAC function with MacKey as the key and
MacData as the input data

2. J-PAKE over Finite Field

2.1. Protocol Setup

When implemented over a finite field, J-PAKE may use the same group
parameters as DSA [FIPS186-4]. Let p and q be two large primes such
that q | p-1. Let Gq denote a subgroup of Zp* with prime order q.
Let g be a generator for Gq. Any non-identity element in Gq can be a
generator. The two communicating parties, Alice and Bob, both agree
on (p, q, g), which can be hard-wired in the software code. They can
also use the method in NIST FIPS 186-4, Appendix A [FIPS186-4] to
generate (p, q, g). Here, DSA group parameters are used only as an
example. Other multiplicative groups suitable for cryptography can
also be used for the implementation, e.g., groups defined in
[RFC4419]. A group setting that provides 128-bit security or above
is recommended. The security proof of J-PAKE depends on the
Decisional Diffie-Hellman (DDH) problem being intractable in the
considered group.

Let s be a secret value derived from a low-entropy password shared
between Alice and Bob. The value of s is REQUIRED to fall within the
range of [1, q-1]. (Note that s must not be 0 for any non-empty
secret.) This range is defined as a necessary condition in [HR08]
for proving the "on-line dictionary attack resistance", since s, s+q,
s+2q, ..., are all considered equivalent values as far as the
protocol specification is concerned. In a practical implementation,
one may obtain s by taking a cryptographic hash of the password and
wrapping the result with respect to modulo q. Alternatively, one may
simply treat the password as an octet string and convert the string
to an integer modulo q by following the method defined in
Section 2.3.8 of [SEC1]. In either case, one MUST ensure s is not
equal to 0 modulo q.

In this round, the sender must send zero knowledge proofs to
demonstrate the knowledge of the ephemeral private keys. A suitable
technique is to use the Schnorr NIZK proof [RFC8235]. As an example,
suppose one wishes to prove the knowledge of the exponent for D = g^d
mod p. The generated Schnorr NIZK proof will contain: {UserID,
V = g^v mod p, r = v - d * c mod q}, where UserID is the unique
identifier for the prover, v is a number chosen uniformly at random
from [0, q-1] and c = H(g || V || D || UserID). The "uniqueness" of
UserID is defined from the user's perspective -- for example, if
Alice communicates with several parties, she shall associate a unique
identity with each party. Upon receiving a Schnorr NIZK proof, Alice
shall check the prover's UserID is a valid identity and is different
from her own identity. During the key exchange process using J-PAKE,
each party shall ensure that the other party has been consistently
using the same identity throughout the protocol execution. Details
about the Schnorr NIZK proof, including the generation and the
verification procedures, can be found in [RFC8235].

When this round finishes, Alice verifies the received ZKPs as
specified in [RFC8235] and also checks that g4 != 1 mod p.
Similarly, Bob verifies the received ZKPs and also checks that
g2 != 1 mod p. If any of these checks fails, this session should be
aborted.

In this round, the Schnorr NIZK proof is computed in the same way as
in the previous round except that the generator is different. For
Alice, the generator used is (g1*g3*g4) instead of g; for Bob, the
generator is (g1*g2*g3) instead of g. Since any non-identity element
in Gq can be used as a generator, Alice and Bob just need to ensure
g1*g3*g4 != 1 mod p and g1*g2*g3 != 1 mod p. With overwhelming
probability, these inequalities are statistically guaranteed even
when the user is communicating with an adversary (i.e., in an active
attack). Nonetheless, for absolute guarantee, the receiving party
shall explicitly check if these inequalities hold, and abort the
session in case such a check fails.

When the second round finishes, Alice and Bob verify the received
ZKPs. If the verification fails, the session is aborted. Otherwise,
the two parties compute the common key material as follows:

Here, Ka = Kb = g^((x1+x3)*x2*x4*s) mod p. Let K denote the same key
material held by both parties. Using K as input, Alice and Bob then
apply a Key Derivation Function (KDF) to derive a common session key
k. If the subsequent secure communication uses a symmetric cipher in
an authenticated mode (say AES-GCM), then one key is sufficient,
i.e., k = KDF(K). Otherwise, the session key should comprise an
encryption key (for confidentiality) and a MAC key (for integrity),
i.e., k = k_enc || k_mac, where k_enc = KDF(K || "JPAKE_ENC") and
k_mac = KDF(K || "JPAKE_MAC"). The exact choice of the KDF is left
to specific applications to define.

2.3. Computational Cost

The computational cost is estimated based on counting the number of
modular exponentiations since they are the predominant cost factors.
Note that it takes one exponentiation to generate a Schnorr NIZK
proof and two to verify it [RFC8235]. For Alice, she needs to
perform 8 exponentiations in the first round, 4 in the second round,
and 2 in the final computation of the session key. Hence, that is 14
modular exponentiations in total. Based on the symmetry, the
computational cost for Bob is exactly the same.

3. J-PAKE over Elliptic Curve

3.1. Protocol Setup

The J-PAKE protocol works basically the same in the elliptic curve
(EC) setting, except that the underlying multiplicative group over a
finite field is replaced by an additive group over an elliptic curve.
Nonetheless, the EC version of J-PAKE is specified here for
completeness.

When implemented over an elliptic curve, J-PAKE may use the same EC
parameters as ECDSA [FIPS186-4]. The FIPS 186-4 standard [FIPS186-4]
defines three types of curves suitable for ECDSA: pseudorandom curves
over prime fields, pseudorandom curves over binary fields, and
special curves over binary fields called Koblitz curves or anomalous
binary curves. All these curves that are suitable for ECDSA can also
be used to implement J-PAKE. However, for illustration purposes,
only curves over prime fields are described in this document.
Typically, such curves include NIST P-256, P-384, and P-521. When
choosing a curve, a level of 128-bit security or above is
recommended. Let E(Fp) be an elliptic curve defined over a finite
field Fp, where p is a large prime. Let G be a generator for the
subgroup over E(Fp) of prime order n. Here, the NIST curves are used
only as an example. Other secure curves such as Curve25519 are also
suitable for implementation. The security proof of J-PAKE relies on
the assumption that the DDH problem is intractable in the considered
group.

As before, let s denote the shared secret between Alice and Bob. The
value of s falls within [1, n-1]. In particular, note that s MUST
not be equal to 0 mod n.

When this round finishes, Alice and Bob verify the received ZKPs as
specified in [RFC8235]. As an example, to prove the knowledge of the
discrete logarithm of D = G x [d] with respect to the base point G,
the ZKP contains: {UserID, V = G x [v], r = v - d * c mod n}, where
UserID is the unique identifier for the prover, v is a number chosen
uniformly at random from [1, n-1] and c = H(G || V || D || UserID).

The verifier shall check the prover's UserID is a valid identity and
is different from its own identity. If the verification of the ZKP
fails, the session is aborted.

When the second round finishes, Alice and Bob verify the received
ZKPs. The ZKPs are computed in the same way as in the previous round
except that the generator is different. For Alice, the new generator
is G1 + G3 + G4; for Bob, it is G1 + G2 + G3. Alice and Bob shall
check that these new generators are not points at infinity. If any
of these checks fails, the session is aborted. Otherwise, the two
parties compute the common key material as follows:

Here, Ka = Kb = G x [(x1+x3)*(x2*x4*s)]. Let K denote the same key
material held by both parties. Using K as input, Alice and Bob then
apply a Key Derivation Function (KDF) to derive a common session key
k.

3.3. Computational Cost

In the EC setting, the computational cost of J-PAKE is estimated
based on counting the number of scalar multiplications over the
elliptic curve. Note that it takes one multiplication to generate a
Schnorr NIZK proof and one to verify it [RFC8235]. For Alice, she
has to perform 6 multiplications in the first round, 3 in the second
round, and 2 in the final computation of the session key. Hence,
that is 11 multiplications in total. Based on the symmetry, the
computational cost for Bob is exactly the same.

4. Three-Pass Variant

The two-round J-PAKE protocol is completely symmetric, which
significantly simplifies the security analysis. In practice, one
party normally initiates the communication and the other party
responds. In that case, the protocol will be completed in three
passes instead of two rounds. The two-round J-PAKE protocol can be
trivially changed to three passes without losing security. Take the
finite field setting as an example, and assume Alice initiates the
key exchange. The three-pass variant works as follows:

Both parties compute the session keys in exactly the same way as
before.

5. Key Confirmation

The two-round J-PAKE protocol (or the three-pass variant) provides
cryptographic guarantee that only the authenticated party who used
the same password at the other end is able to compute the same
session key. So far, the authentication is only implicit. The key
confirmation is also implicit [Stinson06]. The two parties may use
the derived key straight away to start secure communication by
encrypting messages in an authenticated mode. Only the party with
the same derived session key will be able to decrypt and read those
messages.

For achieving explicit authentication, an additional key confirmation
procedure should be performed. This provides explicit assurance that
the other party has actually derived the same key. In this case, the
key confirmation is explicit [Stinson06].

In J-PAKE, explicit key confirmation is recommended whenever the
network bandwidth allows it. It has the benefit of providing
explicit and immediate confirmation if the two parties have derived
the same key and hence are authenticated to each other. This allows
a practical implementation of J-PAKE to effectively detect online
dictionary attacks (if any), and stop them accordingly by setting a
threshold for the consecutively failed connection attempts.

To achieve explicit key confirmation, there are several methods
available. They are generically applicable to all key exchange
protocols, not just J-PAKE. In general, it is recommended that a
different key from the session key be used for key confirmation --
say, k' = KDF(K || "JPAKE_KC"). The advantage of using a different
key for key confirmation is that the session key remains
indistinguishable from random after the key confirmation process.
(However, this perceived advantage is actually subtle and only
theoretical.) Two explicit key confirmation methods are presented
here.

The first method is based on the one used in the SPEKE protocol
[Jab96]. Suppose Alice initiates the key confirmation. Alice sends
to Bob H(H(k')), which Bob will verify. If the verification is
successful, Bob sends back to Alice H(k'), which Alice will verify.
This key confirmation procedure needs to be completed in two rounds,
as shown below.

1. Alice -> Bob: H(H(k'))
2. Bob -> Alice: H(k')

The above procedure requires two rounds instead of one, because the
second message depends on the first. If both parties attempt to send
the first message at the same time without an agreed order, they
cannot tell if the message that they receive is a genuine challenge
or a replayed message, and consequently may enter a deadlock.

The second method is based on the unilateral key confirmation scheme
specified in NIST SP 800-56A Revision 1 [BJS07]. Alice and Bob send
to each other a MAC tag, which they will verify accordingly. This
key confirmation procedure can be completed in one round.

The second method assumes an additional secure MAC function (e.g.,
one may use HMAC) and is slightly more complex than the first method.
However, it can be completed within one round and it preserves the
overall symmetry of the protocol implementation. For this reason,
the second method is RECOMMENDED.

6. Security Considerations

A PAKE protocol is designed to provide two functions in one protocol
execution. The first one is to provide zero-knowledge authentication
of a password. It is called "zero knowledge" because at the end of
the protocol, the two communicating parties will learn nothing more
than one bit information: whether the passwords supplied at two ends
are equal. Therefore, a PAKE protocol is naturally resistant against
phishing attacks. The second function is to provide session key
establishment if the two passwords are equal. The session key will
be used to protect the confidentiality and integrity of the
subsequent communication.

More concretely, a secure PAKE protocol shall satisfy the following
security requirements [HR10].

Offline dictionary attack resistance: It does not leak any
information that allows a passive/active attacker to perform
offline exhaustive search of the password.

Forward secrecy: It produces session keys that remain secure even
when the password is later disclosed.

Known-key security: It prevents a disclosed session key from
affecting the security of other sessions.

Online dictionary attack resistance: It limits an active attacker
to test only one password per protocol execution.

First, a PAKE protocol must resist offline dictionary attacks. A
password is inherently weak. Typically, it has only about 20-30 bits
entropy. This level of security is subject to exhaustive search.
Therefore, in the PAKE protocol, the communication must not reveal
any data that allows an attacker to learn the password through
offline exhaustive search.

Second, a PAKE protocol must provide forward secrecy. The key
exchange is authenticated based on a shared password. However, there
is no guarantee on the long-term secrecy of the password. A secure
PAKE scheme shall protect past session keys even when the password is
later disclosed. This property also implies that if an attacker
knows the password but only passively observes the key exchange, he
cannot learn the session key.

Third, a PAKE protocol must provide known key security. A session
key lasts throughout the session. An exposed session key must not
cause any global impact on the system, affecting the security of
other sessions.

Finally, a PAKE protocol must resist online dictionary attacks. If
the attacker is directly engaging in the key exchange, there is no
way to prevent such an attacker trying a random guess of the
password. However, a secure PAKE scheme should minimize the effect
of the online attack. In the best case, the attacker can only guess
exactly one password per impersonation attempt. Consecutively failed
attempts can be easily detected, and the subsequent attempts shall be
thwarted accordingly. It is recommended that the false
authentication counter be handled in such a way that any error (which
causes the session to fail during the key exchange or key
confirmation) leads to incrementing the false authentication counter.

It has been proven in [HR10] that J-PAKE satisfies all of the four
requirements based on the assumptions that the Decisional Diffie-
Hellman problem is intractable and the underlying Schnorr NIZK proof
is secure. An independent study that proves security of J-PAKE in a
model with algebraic adversaries and random oracles can be found in
[ABM15]. By comparison, it has been known that EKE has the problem
of leaking partial information about the password to a passive
attacker, hence not satisfying the first requirement [Jas96]. For
SPEKE and SRP-6, an attacker may be able to test more than one
password in one online dictionary attack (see [Zha04] and [Hao10]),
hence they do not satisfy the fourth requirement in the strict
theoretical sense. Furthermore, SPEKE is found vulnerable to an
impersonation attack and a key-malleability attack [HS14]. These two
attacks affect the SPEKE protocol specified in Jablon's original 1996
paper [Jab96] as well in the D26 draft of IEEE P1363.2 and the ISO/
IEC 11770-4:2006 standard. As a result, the specification of SPEKE
in ISO/IEC 11770-4:2006 has been revised to address the identified
problems.

Acknowledgements

The editor would like to thank Dylan Clarke, Siamak Shahandashti,
Robert Cragie, Stanislav Smyshlyaev, and Russ Housley for many useful
comments. This work is supported by EPSRC First Grant (EP/J011541/1)
and ERC Starting Grant (No. 306994).

Author's Address

Feng Hao (editor)
Newcastle University (UK)
Urban Sciences Building, School of Computing, Newcastle University
Newcastle Upon Tyne
United Kingdom